Aliquot sum
Last Updated :
22 Jun, 2022
In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself.
They are defined by the sums of their aliquot divisors. The aliquot divisors of a number are all of its divisors except the number itself. The aliquot sum is the sum of the aliquot divisors so, for example, the aliquot divisors of 12 are 1, 2, 3, 4, and 6 and it’s aliquot sum is 16.
A number whose aliquot sum equals its value is a PERFECT number (6 for example).
Examples :
Input : 12
Output : 16
Explanation :
Proper divisors of 12 is = 1, 2, 3, 4, 6
and sum 1 + 2 + 3 + 4 + 6 = 16
Input : 15
Output : 9
Explanation :
Proper divisors of 15 is 1, 3, 5
and sum 1 + 3 + 5 = 9
A simple solution is to traverse through all numbers smaller than n. For every number i, check if i divides n. If yes, we add it to result.
C++
#include <iostream>
using namespace std;
int aliquotSum( int n)
{
int sum = 0;
for ( int i = 1; i < n; i++)
if (n % i == 0)
sum += i;
return sum;
}
int main()
{
int n = 12;
cout << aliquotSum(n);
return 0;
}
|
Java
import java.io.*;
class GFG {
static int aliquotSum( int n)
{
int sum = 0 ;
for ( int i = 1 ; i < n; i++)
if (n % i == 0 )
sum += i;
return sum;
}
public static void main(String args[])
throws IOException
{
int n = 12 ;
System.out.println(aliquotSum(n));
}
}
|
Python3
def aliquotSum(n) :
sm = 0
for i in range ( 1 ,n) :
if (n % i = = 0 ) :
sm = sm + i
return sm
n = 12
print (aliquotSum(n))
|
C#
using System;
class GFG {
static int aliquotSum( int n)
{
int sum = 0;
for ( int i = 1; i < n; i++)
if (n % i == 0)
sum += i;
return sum;
}
public static void Main()
{
int n = 12;
Console.WriteLine(aliquotSum(n));
}
}
|
PHP
<?php
function aliquotSum( $n )
{
$sum = 0;
for ( $i = 1; $i < $n ; $i ++)
if ( $n % $i == 0)
$sum += $i ;
return $sum ;
}
$n = 12;
echo (aliquotSum( $n ));
?>
|
Javascript
<script>
function aliquotSum(n)
{
let sum = 0;
for (let i = 1; i < n; i++)
if (n % i == 0)
sum += i;
return sum;
}
let n = 12;
document.write(aliquotSum(n));
</script>
|
Output :
16
Time Complexity: O(n)
Auxiliary Space: O(1)
Efficient Solutions :
Sum of all proper divisors of a natural number
Sum of all the factors of a number
Please suggest if someone has a better solution which is more efficient in terms of space and time.
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